# IV-estimation vs. Heckman's selection model

I am trying to grasp the difference between IV-estimation and Heckman's selection model.

I do that by considering the following set-up. My outcome if interest is y (I observe it for all observations, incidental truncation is not the problem here!). I have an endogenous binary treatment variable w. Also, I have a binary control function x. Finally, I have an instrument z for my binary treatment variable w. I know how to use instrumental variables, but I have a hard time understanding Heckman's selection model in this case. The first stage should be a probit model, where I regress w on x and z. But I don't understand how my second stage is defined and which assumptions it relies on. I would like to know how to estimate both the average treatment effect and the average treatment effect on the treated. And also how the exclusion restriction kicks in (including or not including z in the second stage).

Hope that someone can provide intuition.

• You might find usefull Heckman, Urzua, and Vytlacil .(2006). Understanding instrumental variables in models with essential heterogeneity. The Review of Economics and Statistics. – José Gabriel Astaiza-Gómez Sep 16 '19 at 14:52
• What do you mean by "binary control function"? – Durden Sep 4 '20 at 17:57

## 1 Answer

1)The intuition of Heckman's model: there are two states of the world and a cost of choosing one of the states. The outcome variable in each state and the cost function have unobservables (error terms). The researcher only observes the data of one of the states, but cannot observe both states for the same unit.

2) The procedure: a) estimate a propensity score for each group of the data i.e. treated and untreated units. Under the assumption of joint normality of the unobservables, use a probit. Without curvature assumptions, use non-parametric estimation. With normality assumptions, you do not need an exclusion restriction (you may need it because of economic arguments). To use non-parametric estimation, you need the exclusion restrictions. b) For each group of variables, regress the outcome variables against the observables and the correction terms. With the abovementioned normality assumptions, those terms are the inverse Mills ratios which are a function of the propensity scores.

3)Average Treatment Effect: use the difference between the parameters of the outcome equations of the second step.

You might find usefull Heckman, Urzua, and Vytlacil .(2006). Understanding instrumental variables in models with essential heterogeneity. The Review of Economics and Statistics.